I noticed that the rules say it must be a F2P account; if a paying member goes into a F2P server, they have no benefits of P2P and are therefore, for all practical purposes, running as F2P.

Also, if it's a quest I've done before on that account, I can still redo the quest and submit that, right? Or I'd need a new account every time I make an attempt. ?

Has to be a new account. Can you even redo quests on RS3?

All of our current categories either require starting from a fresh account or using a members account to redo a minigame/quest. So no, using an existing members account does not count for Dragon Slayer. If you have any ideas for other categories feel free to try them!